Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint

Beniamin Bogosel (1) , Edouard Oudet (2)

1 LAMA - Laboratoire de Mathématiques
2 CVGI - Calcul des Variations, Géométrie, Image

We consider the problem of optimizing the k th eigenvalue of the Dirichlet Laplace operator under perimeter constraint. We provide a new method based on a Γ-convergence result for approximating the corresponding optimal shapes. We also give new optimality conditions in the case of multiple eigenvalues. We deduce from previous conditions the fact that optimal shapes never contain flat parts in their boundaries.

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Beniamin Bogosel, Edouard Oudet. Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint. SIAM Journal on Control and Optimization, 2016, 54 (1), pp.317-340. ⟨10.1137/140999530⟩. ⟨hal-02009510⟩